Positive solutions of a forced nonlinear elliptic boundary value problem (Q1964217)

Let \(D\) be a bounded domain in \(\mathbb R^n\) with \(C^\infty\) boundary and define the operators \(A\) and \(B\) by \[ Au= - D_i(a^{ij}D_j u) + cu,\quad Bu=a a^{ij}D_iu\gamma_j +(1-a)u, \] where \([a_{ij}]\) is a positive-definite matrix, \(\gamma\) is the unit exterior normal to \(\partial D\), and \(a^{ij}\), \(c\), and \(a\) are \(C^\infty\) functions with \(0\leq a \leq 1\) and \(c > 0\). The author is interested in the behavior of solutions of the boundary value problems \[ Au=\lambda u + f(u)\text{ in } D,\;Bu=0 \text{ on } \partial D, \tag{1} \] and \[ Au=\lambda f(u) \text{ in } D, \;Bu=0 \text{ on } \partial D \tag{2} \] with respect to the parameter \(\lambda\) when \(f\) is a \(C^1\) function defined on \([0,\infty)\) such that \(f(0) >0\) and the function \(g\) defined by \(g(t)=f(t)/t\) has a negative derivative on \((0,\infty)\). To describe the behavior of the solutions, we write \(\lambda_1\) for the first eigenvalue of the eigenvalue problem \(Au=\lambda u\) in \(D\), \(Bu=0\) on \(\partial D\), and we set \(k_\infty = \lim_{t\to\infty} g(t)\). In a previous paper, the author and \textit{K. Taira} [Nonlinear Anal., Theory Methods Appl. 29, No. 7, 761-771 (1997; Zbl 0878.35048)] showed that problem (1) has a unique positive solution if and only if \(\lambda < \lambda_1- k_\infty\). In this paper, he shows that the solutions depend in a \(C^1\) fashion on \(\lambda\). In addition, the solutions converge uniformly to \(0\) as \(\lambda\to -\infty\) with \(\|u\|_\infty = O(1/|\lambda|)\) and their maxima tend to infinity as \(\lambda\to \lambda_1-k_\infty\). For problem (2), there are two cases: if \(k^\infty \geq 0\), there is a positive solution if and only if \(<\lambda< \lambda_1/k_\infty\), and the solutions converge in \(C^{2+\alpha}\) (for any \(\alpha\in (0,1)\)) to zero as \(\lambda\to 0\) while the maxima tend to infinity as \(\lambda\to\lambda_1/k_\infty\). If \(k_\infty <0\), then the solutions converge to \(0\) in \(C^{2+\alpha}\) as \(\lambda \to 0\) while the maxima tend to the first positive zero of \(f\) as \(\lambda\to\infty\). In either case, the solutions still depend on \(\lambda\) in a \(C^1\) fashion. The proofs involve a clever combination of the implicit function theorem, the method of sub- and supersolutions, and a priori estimates.